Problem Set 1---Due February 4
1) Consider
a quantum system composed of two subsystems. Each subsystem is expressed in
terms of a basis of orthonormal states which are eigenstates of some hermitian
operator (associated with some physical properties). A basis for the combined
system can be given by states of the form 
where the first label denotes system
1 and the second system 2. Suppose the state of the system is given by 
.
a) Show that the state is properly normalized.
b) Suppose
both system 1 and 2 are measured. What is the probability that we will find
the system in the state 
?
c) Suppose
system 2 is measured and found to be in state 
. Given this, what is the probability
that system 1 will be in state 
?
d) Suppose
system 2 is measured and found to be in state 
. Given this, what is the probability
that system 1 will be in state ?
2) In
class we considered the two dimensional isotropic harmonic oscillator (where
isotropic means the spring constant is the same in all directions. We showed
the energy spectrum was given 
with the Nth level having
degeneracy of (N+1). In this problem, generalize the analysis to three
dimensions: show that for a three dimensional isotropic harmonic oscillator the
showed the energy spectrum was given 
with the Nth level having
degeneracy of 
.
3) Consider
two distinct systems with two degrees of freedom. The first one is a system
of two particles (of masses 
and 
) confined to a one-dimensional box of
length L. The second is a single particle (of mass m) confined
to a two-dimensional box with sides 
and 
. Show that the two systems have
exactly the same energy spectrum if 
and 
.
4) In
class we discussed the separation of variables for the three dimensional
time-independent Schrödinger equation with a spherically symmetric potential.
In this problem I want you to work thorough the equivalent for a two dimension
system with an axially symmetric potential. In particular consider the equation
in polar
coordinates. Show that the it has solutions of the form 
with 
(integer m) and R
satisfying 
.